Generalized wave polynomials and transmutations related to perturbed Bessel equations

TL;DR

This paper introduces generalized wave polynomials to approximate the integral kernel of the transmutation operator for perturbed Bessel equations, enabling uniform approximation of solutions with spectral-independent estimates.

Contribution

It develops a novel method using generalized wave polynomials to approximate transmutation kernels for perturbed Bessel equations, providing spectral-independent error estimates.

Findings

Kernel approximated uniformly by generalized wave polynomials

Approximate solutions with estimates independent of spectral parameter

New approach for solving perturbed Bessel equations

Abstract

The transmutation (transformation) operator associated with the perturbed Bessel equation is considered. It is shown that its integral kernel can be uniformly approximated by linear combinations of constructed here generalized wave polynomials, solutions of a singular hyperbolic partial differential equation arising in relation with the transmutation kernel. As a corollary of this results an approximation of the regular solution of the perturbed Bessel equation is proposed with corresponding estimates independent of the spectral parameter.